Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-inf2vnlem3 Structured version   GIF version

Theorem bj-inf2vnlem3 9402
Description: Lemma for bj-inf2vn 9404. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-inf2vnlem3.bd1 BOUNDED A
bj-inf2vnlem3.bd2 BOUNDED 𝑍
Assertion
Ref Expression
bj-inf2vnlem3 (x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
Distinct variable groups:   x,y,A   x,𝑍,y

Proof of Theorem bj-inf2vnlem3
Dummy variables z 𝑡 u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 9401 . . 3 (x A (x = ∅ y A x = suc y) → (Ind 𝑍u(𝑡 u (𝑡 A𝑡 𝑍) → (u Au 𝑍))))
2 bj-inf2vnlem3.bd1 . . . . . 6 BOUNDED A
32bdeli 9281 . . . . 5 BOUNDED z A
4 bj-inf2vnlem3.bd2 . . . . . 6 BOUNDED 𝑍
54bdeli 9281 . . . . 5 BOUNDED z 𝑍
63, 5ax-bdim 9249 . . . 4 BOUNDED (z Az 𝑍)
7 nfv 1418 . . . 4 z(𝑡 A𝑡 𝑍)
8 nfv 1418 . . . 4 z(u Au 𝑍)
9 nfv 1418 . . . 4 u(z Az 𝑍)
10 nfv 1418 . . . 4 u(𝑡 A𝑡 𝑍)
11 eleq1 2097 . . . . . 6 (z = 𝑡 → (z A𝑡 A))
12 eleq1 2097 . . . . . 6 (z = 𝑡 → (z 𝑍𝑡 𝑍))
1311, 12imbi12d 223 . . . . 5 (z = 𝑡 → ((z Az 𝑍) ↔ (𝑡 A𝑡 𝑍)))
1413biimpd 132 . . . 4 (z = 𝑡 → ((z Az 𝑍) → (𝑡 A𝑡 𝑍)))
15 eleq1 2097 . . . . . 6 (z = u → (z Au A))
16 eleq1 2097 . . . . . 6 (z = u → (z 𝑍u 𝑍))
1715, 16imbi12d 223 . . . . 5 (z = u → ((z Az 𝑍) ↔ (u Au 𝑍)))
1817biimprd 147 . . . 4 (z = u → ((u Au 𝑍) → (z Az 𝑍)))
196, 7, 8, 9, 10, 14, 18bdsetindis 9399 . . 3 (u(𝑡 u (𝑡 A𝑡 𝑍) → (u Au 𝑍)) → z(z Az 𝑍))
201, 19syl6 29 . 2 (x A (x = ∅ y A x = suc y) → (Ind 𝑍z(z Az 𝑍)))
21 dfss2 2928 . 2 (A𝑍z(z Az 𝑍))
2220, 21syl6ibr 151 1 (x A (x = ∅ y A x = suc y) → (Ind 𝑍A𝑍))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628  wal 1240   = wceq 1242   wcel 1390  wral 2300  wrex 2301  wss 2911  c0 3218  suc csuc 4068  BOUNDED wbdc 9275  Ind wind 9361
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-bdim 9249  ax-bdsetind 9398
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-suc 4074  df-bdc 9276  df-bj-ind 9362
This theorem is referenced by:  bj-inf2vn  9404
  Copyright terms: Public domain W3C validator